Physics

John D. Cutnell, Kenneth W. Johnson

Chapter 21

Magnetic Forces and Magnetic Fields - all with Video Answers

Educators

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Chapter Questions

Problem 1

ssm In New England, the horizontal component of the earth's magnetic field has a magnitude of $1.6 \times 10^{-5}$ T. An electron is shot vertically straight up from the ground with a speed of $2.1 \times 10^{6} \mathrm{m} / \mathrm{s}$ . What is the magnitude of the acceleration caused by the magnetic force? Ignore the gravitational force acting on the electron.

Jarrad Pond

Jarrad Pond

Numerade Educator

Problem 2

(a) A proton, traveling with a velocity of $4.5 \times 10^{6} \mathrm{m} / \mathrm{s}$ due east, experiences a magnetic force that has a maximum magnitude of $8.0 \times 10^{-14} \mathrm{N}$ and a direction of due south. What are the magnitude and direction of the magnetic field causing the force? (b) Repeat part (a) assuming the proton is replaced by an electron.

BM

Bob Mckinnon

Numerade Educator

Problem 3

ssm At a certain location, the horizontal component of the earth's magnetic field is $2.5 \times 10^{-5} \mathrm{T}$ , due north. A proton moves eastward with just the right speed for the magnetic force on it to balance its weight. Find the speed of the proton.

Jarrad Pond

Numerade Educator

Problem 4

A charge of $-8.3 \mu \mathrm{C}$ is traveling at a speed of $7.4 \times 10^{6} \mathrm{m} / \mathrm{s}$ in a region of space where there is a magnetic field. The angle between the velocity of the charge and the field is $52^{\circ} .$ A force of magnitude $5.4 \times 10^{-3} \mathrm{N}$ acts on the charge. What is the magnitude of the magnetic field?

BM

Bob Mckinnon

Numerade Educator

Problem 5

When a charged particle moves at an angle of $25^{\circ}$ with respect to a magnetic field, it experiences a magnetic force of magnitude $F$ At what angle (less than $90^{\circ}$ ) with respect to this field will this particle, moving at the same speed, experience a magnetic force of magnitude 2$F ?$

Jarrad Pond

Numerade Educator

Problem 6

A particle that has an 8.2$\mu \mathrm{C}$ charge moves with a velocity of magnitude $5.0 \times 10^{5} \mathrm{m} / \mathrm{s}$ along the $+x$ axis. It experiences no magnetic force, although there is a magnetic field present. The maximum possible magnetic force that the charge could experience has a magnitude of 0.48 $\mathrm{N}$ . Find the magnitude and direction of the magnetic field. Note that there are two possible answers for the direction of the field.

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Bob Mckinnon

Numerade Educator

Problem 7

A magnetic field has a magnitude of $1.2 \times 10^{-3} \mathrm{T}$ , and an electric field has a magnitude of $4.6 \times 10^{3} \mathrm{N} / \mathrm{C}$ . Both fields point in the same direction. A positive 1.8$\mu \mathrm{C}$ charge moves at a speed of $3.1 \times 10^{6} \mathrm{m} / \mathrm{s}$ in a direction that is perpendicular to both fields. Determine the magnitude of the net force that acts on the charge.

Jarrad Pond

Numerade Educator

Problem 8

Two charged particles move in the same direction with respect to the same magnetic field. Particle 1 travels three times faster than particle 2. However, each particle experiences a magnetic force of the same magnitude. Find the ratio $\left|q_{1}\right| /\left|q_{2}\right|$ of the magnitudes of the charges.

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Bob Mckinnon

Numerade Educator

Problem 9

The drawing shows a parallel plate capacitor that is moving with a speed of 32 $\mathrm{m} / \mathrm{s}$ through a $3.6-\mathrm{T}$ magnetic field. The velocity $\overrightarrow{\mathbf{v}}$ is perpendicular to the magnetic field. The electric field within the capacitor has a value of $170 \mathrm{N} / \mathrm{C},$ and each plate has an area of $7.5 \times 10^{-4} \mathrm{m}^{2}$ . What is the magnetic force (magnitude and direction) exerted on the positive plate of the capacitor?

Jarrad Pond

Numerade Educator

Problem 10

One component of a magnetic field has a magnitude of 0.048 $\mathrm{T}$ and points along the $+x$ axis, while the other component has a magnitude of 0.065 $\mathrm{T}$ and points along the $-y$ axis. A particle carrying a charge of $+2.0 \times 10^{-5} \mathrm{C}$ is moving along the $+z$ axis at a speed of $4.2 \times 10^{3} \mathrm{m} / \mathrm{s} .$ (a) Find the magnitude of the net magnetic force that acts on the particle. (b) Determine the angle that the net force makes with respect to the $+x$ axis.

BM

Bob Mckinnon

Numerade Educator

Problem 11

ssm The electrons in the beam of a television tube have a kinetic energy of $2.40 \times 10^{-15} \mathrm{J}$ . Initially, the clectrons move horizontally from west to east. The vertical component of the earth's magnetic field points down, toward the surface of the earth, and has a magnitude of $2.00 \times 10^{-5} \mathrm{T} .$ (a) In what direction are the electrons deflected by this field component? (b) What is the acceleration of an electron in part (a)?

Jarrad Pond

Numerade Educator

Problem 12

An ionized helium atom has a mass of $6.6 \times 10^{-27} \mathrm{kg}$ and a speed of $4.4 \times 10^{5} \mathrm{m} / \mathrm{s}$ . It moves perpendicular to a $0.75-\mathrm{T}$ magnetic field on a circular path that has a 0.012 -m radius. Determine whether the charge of the ionized atom is $+e$ or $+2 e .$

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Bob Mckinnon

Numerade Educator

Problem 13

ssm In the operating room, anesthesiologists use mass spectrometers to monitor the respiratory gases of patients undergoing surgery. One gas that is often monitored is the anesthetic isoflurane (molecular mass $=3.06 \times 10^{-25} \mathrm{kg} ) .$ In a spectrometer, a singly ionized molecule of isoflurane (charge $=+e )$ moves at a speed of $7.2 \times 10^{3} \mathrm{m} / \mathrm{s}$ /s on a circular path that hat has a radius of 0.10 $\mathrm{m}$ . What is the magnitude of the magnetic field that the spectrometer uses?

Jarrad Pond

Numerade Educator

Problem 14

A charged particle with a charge-to-mass ratio of $|q| / m=5.7 \times 10^{8} \mathrm{C} / \mathrm{kg}$ travels on a circular path that is perpendicular to a magnetic field whose magnitude is 0.72 $\mathrm{T}$ . How much time does it take for the particle to complete one revolution?

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Bob Mckinnon

Numerade Educator

Problem 15

A charged particle enters a uniform magnetic field and follows the circular path shown in the drawing. (a) Is the particle positively or negatively charged? Why? (b) The particle’s speed is 140 $\mathrm{m} / \mathrm{s}$ , the magnitude of the magnetic field is $0.48 \mathrm{T},$ and the radius of the path is 960 $\mathrm{m}$ . Determine the mass of the particle, given that its charge has a magnitude of $8.2 \times 10^{-4} \mathrm{C}$

Jarrad Pond

Numerade Educator

Problem 16

A proton is projected perpendicularly into a magnetic field that has a magnitude of 0.50 T. The ficld is then adjusted so that an electron will follow a circular path of the same radius when it is projected perpendicularly into the field with the same velocity that the proton had. What is the magnitude of the field used for the electron?

BM

Bob Mckinnon

Numerade Educator

Problem 17

ssm When beryllium-7 ions $\left(m=11.65 \times 10^{-27} \mathrm{kg}\right)$ pass through a mass spectrometer, a uniform magnetic field of 0.283 $\mathrm{T}$ curves their path directly to the center of the detector (see Figure 21.14$) .$ For the same accelerating potential difference, what magnetic field should be used to send beryllium-10 ions $\left(m=16.63 \times 10^{-27} \mathrm{kg}\right)$ to the same location in the detector? Both types of ions are singly ionized $(q=+e)$ .

Jarrad Pond

Numerade Educator

Problem 18

Suppose that an ion source in a mass spectrometer produces doubly ionized gold ions (Au $^{2+}$ ), each with a mass of $3.27 \times 10^{-25} \mathrm{kg}$ . The ions are accelerated from rest through a potential difference of 1.00 $\mathrm{kV}$ . Then, a $0.500-\mathrm{T}$ magnetic field causes the ions to follow a circular path. Determine the radius of the path.

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Bob Mckinnon

Numerade Educator

Problem 19

An $\alpha$ -particle has a charge of $+2 e$ and a mass of $6.64 \times 10^{-27} \mathrm{kg}$ . It is accelerated from rest through a potential difference that has a value of $1.20 \times 10^{6} \mathrm{V}$ and then enters a uniform magnetic field whose magnitude is 2.20 $\mathrm{T}$ . The $\alpha$ -particle moves perpendicular to the magnetic field at all times. What is $(\mathrm{a})$ the speed of the $\alpha$ -particle, (b) the magnitude of the magnetic force on it, and (c) the radius of its circular path?

Jarrad Pond

Numerade Educator

Problem 20

Particle 1 and particle 2 have masses of $m_{1}=2.3 \times 10^{-8} \mathrm{kg}$ and $m_{2}=5.9 \times 10^{-8} \mathrm{kg},$ but they carry the same charge $q .$ The two particles accelerate from rest through the same electric potential difference $V$ and enter the same magnetic field, which has a magnitude $B$. The particles travel perpendicular to the magnetic field on circular paths. The radius of the circular path for particle 1 is $r_{1}=12 \mathrm{cm} .$ What is the radius (in cm) of the circular path for particle 2$?$

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Bob Mckinnon

Numerade Educator

Problem 21

Two of the isotopes of cartoon, carbon-12 and carbon-13, have masses of $19.93 \times 10^{-27} \mathrm{kg}$ and $21.59 \times 10^{-27} \mathrm{kg}$ , respectively. These two isotopes are singly ionized $(+e),$ each given a speed of $6.667 \times 10^{5} \mathrm{m} / \mathrm{s}$ . The ions then enter the bending region of a mass spectrometer where the magnetic field is 0.8500 $\mathrm{T}$ . Determine the spatial separation between the two isotopes after they have traveled through a half-circle.

Jarrad Pond

Numerade Educator

Problem 22

The ion source in a mass spectrometer produces both singly and doubly ionized species, $\mathrm{X}^{+}$ and $\mathrm{X}^{2+}$ . The difference in mass between these species is too small to be detected. Both species are accelerated through the same electric potential difference, and both experience the same magnetic ficld, which causes them to move on circular paths. The radius of the path for the species $X^{+}$ is $r_{1},$ while the radius for species $X^{2+}$ is $r_{2}$ . Find the ratio $r_{1} / r_{2}$ of the radii.

BM

Bob Mckinnon

Numerade Educator

Problem 23

ssm mmh A proton with a speed of $3.5 \times 10^{6} \mathrm{m} / \mathrm{s}$ is shot into a region between two plates that are scparated by a distance of 0.23 $\mathrm{m}$ . As the drawing shows, a magnetic field exists between the plates, and it is perpendicular to the velocity of the proton. What must be the magnitude of the magnetic field so the proton just misses colliding with the opposite plate?

Jarrad Pond

Numerade Educator

Problem 24

Review Conceptual Example 2 as an aid in understanding this problem. A velocity selector has an electric field of magnitude $2470 \mathrm{N} / \mathrm{C},$ directed vertically upward, and a horizontal magnetic field that is directed south. Charged particles, traveling east at a speed of $6.50 \times 10^{3} \mathrm{m} / \mathrm{s},$ enter the velocity selector and are able to pass completely through without being deflected. When a different particle with an electric charge of $+4.00 \times 10^{-12} \mathrm{C}$ enters the velocity selector traveling an east, the net force (due to the electric and magnetic fields) acting on it is $1.90 \times 10^{-9} \mathrm{N},$ pointing directly upward. What is the speed of this particle?

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Bob Mckinnon

Numerade Educator

Problem 25

ssm A particle of mass $6.0 \times 10^{-8} \mathrm{kg}$ and charge $+7.2 \mu \mathrm{C}$ is traveling due east. It enters perpendicularly a magnetic field whose magnitude is 3.0 $\mathrm{T}$ . After entering the field, the particle completes one-half of a circle and exits the field traveling due west. How much time does the particle spend traveling in the magnetic field?

Jarrad Pond

Numerade Educator

Problem 26

Conceptual Example 4 provides background pertinent to this problem. An electron has a kinetic energy of $2.0 \times 10^{-17} \mathrm{J}$ . It moves on a circular path that is perpendicular to a uniform magnetic field of magnitude $5.3 \times 10^{-5}$ T. Determine the radius of the path.

BM

Bob Mckinnon

Numerade Educator

Problem 27

A positively charged particle of mass $7.2 \times 10^{-8} \mathrm{kg}$ is traveling due east with a speed of 85 $\mathrm{m} / \mathrm{s}$ and enters a $0.31-\mathrm{T}$ uniform magnetic field. The particle moves through one-quarter of a circle in a time of $2.2 \times 10^{-3} \mathrm{s}$ , at which time it leaves the field heading due south. All during the motion the particle moves perpendicular to the magnetic field. (a) What is the magnitude of the magnetic force acting on the particle? (b) Determine the magnitude of its charge.

Jarrad Pond

Numerade Educator

Problem 28

Review Conceptual Example 2 as background for this problem. A charged particle moves through a velocity selector at a constant speed in a straight line. The electric field of the velocity selector is $3.80 \times 10^{3} \mathrm{N} / \mathrm{C}$ , while the magnetic field is 0.360 $\mathrm{T}$ the electric field is turned off, the charged particle travels on a circular path whose radius is 4.30 $\mathrm{cm} .$
Find the charge-to-mass ratio of the particle.

BM

Bob Mckinnon

Numerade Educator

Problem 29

Refer to Check Your Understanding Question 10 before starting this problem. Suppose that the target discussed there is located at the coordinates $x=-0.10 \mathrm{m}$ and $y=-0.10 \mathrm{m} .$ In addition, suppose that the particle is a proton and the magnetic field has a magnitude of 0.010 $\mathrm{T}$ . The speed at which the particle is projected is the same for either of the two paths leading to the target. Find the speed.

Jarrad Pond

Numerade Educator

Problem 30

At New York City, the earth's magnetic field has a vertical component of $5.2 \times 10^{-5} \mathrm{T}$ that points downward (perpendicular to the ground) and a horizontal componcnt of $1.8 \times 10^{-5} \mathrm{T}$ that points toward gcographic north (parallel to the ground). What are the magnitude and direction of the magnetic force on a $6.0-\mathrm{m}$ long, straight wire that carries a current of 28 A perpendicularly into the ground?

BM

Bob Mckinnon

Numerade Educator

Problem 31

ssm A $45-\mathrm{m}$ length of wire is stretched horizontally between two vertical posts. The wire carries a current of 75 $\mathrm{A}$ and experiences a magnetic force of 0.15 $\mathrm{N}$ . Find the magnitude of the earth's magnetic field at the location of the wire, assuming the field makes an angle of $60.0^{\circ}$ with respect to the wire.

Jarrad Pond

Numerade Educator

Problem 32

A straight wire in a magnetic field experiences a force of 0.030 $\mathrm{N}$ . when the current in the wire is 2.7 $\mathrm{A}$ . The current in the wire is changed, and the wire experiences a force of 0.047 $\mathrm{N}$ as a result. What is the new current?

BM

Bob Mckinnon

Numerade Educator

Problem 33

A horizontal wire of length $0.53 \mathrm{m},$ carrying a current of $7.5 \mathrm{A},$ is placed in a uniform external magnetic field. When the wire is horizontal, it experiences no magnetic force. When the wire is tilted upward at an angle of $19^{\circ},$ it experiences a magnetic force of $4.4 \times 10^{-3} \mathrm{N}$ . Determine the magnitude of the external magnetic field.

Jarrad Pond

Numerade Educator

Problem 34

The drawing shows a wire composed of three segments, AB, BC, and CD. There is a current of $I=2.8 \mathrm{A}$ in the wire. There is also a magnetic field $\overrightarrow{\mathrm{B}}$ (magnitude $=0.26 \mathrm{T}$ ) that is the same everywhere and points in the direction of the $+z$ axis. The lengths of the wire segments are $L_{\mathrm{AB}}=1.1 \mathrm{m}, L_{\mathrm{BC}}=0.55 \mathrm{m},$ and $L_{\mathrm{CD}}=0.55 \mathrm{m}$ . Find the magnitude of the force that acts on each segment.

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Bob Mckinnon

Numerade Educator

Problem 35

ssm A wire carries a current of 0.66 $\mathrm{A}$ . This wire makes an angle of $58^{\circ}$ with respect to a magnetic field of magnitude $4.7 \times 10^{-5} \mathrm{T}$ . The wire experiences a magnetic force of magnitude $7.1 \times 10^{-5} \mathrm{N}$ . What is the length of the wire?

Jarrad Pond

Numerade Educator

Problem 36

Two insulated wires, each 2.40 $\mathrm{m}$ long, are taped together to form a two-wire unit that is 2.40 $\mathrm{m}$ . One wire carries a current of 7.00 $\mathrm{A}$ ; the other carries a smaller current I in the opposite direction. The two-wire unit is placed at an angle of $65.0^{\circ}$ relative to a magnetic field whose magnitude is 0.360 $\mathrm{T}$ . The magnitude of the net magnetic force experienced by the two-wire unit is 3.13 $\mathrm{N}$ . What is the current $I$ ?

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Bob Mckinnon

Numerade Educator

Problem 37

A loop of wire has the shape of a right triangle (see the drawing) and carries a current of $I=4.70$ A. A uniform magnetic field is directed parallel to side $A B$ and has a magnitude of 1.80 $\mathrm{T}$ . (a) Find the magnitude and direction of the magnetic force exerted on each side of the magnetic. (b) Determine the magnitude of the net force exerted on the triangle.

Penny Riley

Numerade Educator

Problem 38

A copper rod of length 0.85 $\mathrm{m}$ is lying on a frictionless table (see the drawing). Each end of the rod is attached to a fixed wire by an unstretched spring that has a spring constant of $k=75 \mathrm{N} / \mathrm{m}$ . A magnetic field with a strength of 0.16 $\mathrm{T}$ is oriented perpendicular to the surface of the table. (a) What must be the direction of the current in the copper rod that causes the springs to stretch? (b) If the current is $12 \mathrm{A},$ by how much does each spring stretch?

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Bob Mckinnon

Numerade Educator

Problem 39

ssm mm The drawing shows a thin, uniform rod that has a length of 0.45 $\mathrm{m}$ and a mass of 0.094 $\mathrm{kg}$ . This rod lies in the plane of the paper and is attached to the floor by a hinge at point $P .$ A uniform magnctic field of 0.36 $\mathrm{T}$ is directed perpendicularly into the plane of the paper. There is a current $I=4.1 \mathrm{A}$ in the rod, which does not rotate clockwise or counterclockwise. Find the angle $\theta$ . (Hint: The magnetic force may be taken to act at the center of gravity.)

Jarrad Pond

Numerade Educator

Problem 40

A horizontal wire is hung from the ceiling of a room by two massless strings. The wire has a length of 0.20 $\mathrm{m}$ and a mass of 0.080 kg. A uniform magnetic field of magnitude 0.070 $\mathrm{T}$ is directed from the ceiling to the floor.When a current of $I=42 \mathrm{A}$ exists in the wire, the wire swings upward and, at equilibrium, makes an angle $\phi$ with respect to the vertical, as the drawing shows. Find (a) the angle $\phi$ and $(b)$ the tension in each of the two strings.

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Bob Mckinnon

Numerade Educator

Problem 41

The two conducting rails in the drawing are tilted upward so they each make an angle of $30.0^{\circ}$ with respect to the ground. The vertical magnetic field has a magnitude of 0.050 T. The $0.20-\mathrm{kg}$ aluminum rod (length $=1.6 \mathrm{m} )$ slides without friction down the rails at a constant velocity. How much current flows through the rod?

Jarrad Pond

Numerade Educator

Problem 42

Two coils have the same number of circular turns and carry the same current. Each rotates in a magnetic field as in Figure 21.19 . Coil 1 has a radius of 5.0 $\mathrm{cm}$ and rotates in a 0.18 -T field. Coil 2 rotates in a $0.42-\mathrm{T}$ field. Each coil experiences the same maximum torque. What is the radius (in $\mathrm{cm} )$ of coil 2$?$

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Bob Mckinnon

Numerade Educator

Problem 43

The $1200-$ -turn coil in a dc motor has an area per turn of $1.1 \times 10^{-2} \mathrm{m}^{2}$ The design for the motor specifies that the magnitude of the maximum torque is 5.8 $\mathrm{N} \cdot \mathrm{m}$ when the coil is placed in a $0.20-\mathrm{T}$ magnetic field What is the current in the coil?

Jarrad Pond

Numerade Educator

Problem 44

Two circular coils of current-carrying wire have the same magnetic moment. The first coil has a radius of $0.088 \mathrm{m},$ has 140 turns, and carries a current of 4.2 $\mathrm{A}$ . The second coil has 170 turns and carries a current of 9.5 $\mathrm{A} .$ What is the radius of the second coil?

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Bob Mckinnon

Numerade Educator

Problem 45

A wire has a length of $7.00 \times 10^{-2} \mathrm{m}$ and is used to make a a circular coil of one turn. There is a current of 4.30 $\mathrm{A}$ in the wire. In the presence of a $2.50-\mathrm{T}$ magnetic field, what is the maximum torque that this coil can experience?

Jarrad Pond

Numerade Educator

Problem 46

The coil of wire in the drawing is a right triangle and is free to rotate about an axis that is attached along side $A C .$ The current in the loop is $I=4.70 \mathrm{A}$ , and the magnetic field (parallel to the plane of the loop and side $A B$ is $B=1.80 \mathrm{T}$ . (a) What is the magnetic moment of the loop, and (b) what is the magnitude of the net torque exerted on the loop by the magnetic field?

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Bob Mckinnon

Numerade Educator

Problem 47

Two pieces of the same wire have the same length. From one piece, a square coil containing a single loop is made. From the other, a circular coil containing a single loop is made. The coils carry different currents. When placed in the same magnetic field with the same orientation, they experience the same torque. What is the ratio $I_{\text { square }} / I_{\text { circle }}$ the current in the square coil to the current in the circular coil?

Jarrad Pond

Numerade Educator

Problem 48

You have a wire of length $L=1.00 \mathrm{m}$ from which to make the square coil of a dc motor. The current in the coil is $I=1.7 \mathrm{A},$ and the magnetic field of the motor has a magnitude of $B=0.34 \mathrm{T}$ . Find the maximum torque exerted on the coil when the wire is used to make a single-turn square coil and a two-turn square coil.

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Bob Mckinnon

Numerade Educator

Problem 49

ssm The rectangular loop in the drawing consists of 75 turns and carries a current of $I=4.4 \mathrm{A}$ . A $1.8-\mathrm{T}$ magnetic field is directed along the $+y$ axis. The loop is free to rotate about the $z$ axis. (a) Determine the magnitude of the net torque exerted on the loop and (b) state whether the $35^{\circ}$ angle will increase or decrease.

Jarrad Pond

Numerade Educator

Problem 50

A square coil and a rectangular coil are each made from the same length of wire. Each contains a single
turn. The long sides of the rectangle are twice as long as the short sides. Find the ratio $\tau_{\text { square }} / \tau_{\text { rectangle }}$ of the maximum torques that these coils experience in the same magnetic field when they contain the same current.

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Bob Mckinnon

Numerade Educator

Problem 51

mmh The coil in Figure 21.19$a$ contains 410 turns and has an area per turn of $3.1 \times 10^{-3} \mathrm{m}^{2}$ . The magnetic field is $0.23 \mathrm{T},$ and the current in the coil is 0.26 $\mathrm{A}$ . A brake shoe is pressed perpendicularly against the shaft to keep the coil from turning. The coefficient of static friction between the shaft and the brake shoe is $0.76 .$ The radius of the shaft is 0.012 m. What is the magnitude of the minimum normal force that the brake shoe exerts on the shaft?

Jarrad Pond

Numerade Educator

Problem 52

In the model of the hydrogen atom created by Niels Bohr, the electron moves around the proton at a speed of $2.2 \times 10^{6} \mathrm{m} / \mathrm{s}$ in a circle of radius $5.3 \times 10^{-11} \mathrm{m}$ . Considering the orbiting electron to be a small current loop, determine the magnetic moment associated with this motion. Hint: The electron travels around the circle in a time equal to the period of the motion.)

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Bob Mckinnon

Numerade Educator

Problem 53

ssm Suppose in Figure 21.27$a$ that $I_{1}=I_{2}=25 \mathrm{A}$ and that the separation between the wires is 0.016 $\mathrm{m} .$ By applying an external magnetic field (created by a source other than the wires) it is possible to cancel the mutual repulsion of the wires. This external field must point along the vertical direction. (a) Does the external field point up or down? Explain. (b) What is the magnitude of the external field?

Jarrad Pond

Numerade Educator

Problem 54

A long solenoid has a length of 0.65 $\mathrm{m}$ and contains 1400 turns of wire. There is a current of 4.7 $\mathrm{A}$ in the wire. What is the magnitude of the magnetic field within the solenoid?

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Bob Mckinnon

Numerade Educator

Problem 55

ssm The magnetic field produced by the solenoid in a magnetic resonance imaging (MRI) system designed for measurements on whole human bodies has a field strength of $7.0 \mathrm{T},$ and the current in the solenoid is 2.0 $\times 10^{2}$ A. What is the number of turns per meter of length of the solenoid? Note that the solenoid used to produce the magnetic field in this type of system has a length that is not very long compared to its diameter. Because of this and other design considerations, your answer will be only an approximation.

Jarrad Pond

Numerade Educator

Problem 56

A long solenoid has 1400 turns per meter of length, and it carries a current of 3.5 $\mathrm{A}$ . A small circular coil of wire is placed inside the solenoid with the normal to the coil oriented at an angle of $90.0^{\circ}$ with respect to the axis of the solenoid. The coil consists of 50 turns, has an area of $1.2 \times 10^{-3} \mathrm{m}^{2}$ , and carries a current of 0.50 A. Find the torque exerted on the coil.

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Bob Mckinnon

Numerade Educator

Problem 57

ssm Two circular loops of wire, each containing a single turn, have the same radius of 4.0 $\mathrm{cm}$ and a common center. The planes of the loops are perpendicular. Each carries a current of 1.7 $\mathrm{A}$ . What is the magnitude of the net magnetic field at the common center?

Jarrad Pond

Numerade Educator

Problem 58

Multiple-Concept Example 8 reviews the concepts from this chapter that are pertinent here. Two rigid rods are oriented parallel to each other and to the ground. The rods carry the same current in the same direction. The length of each rod is $0.85 \mathrm{m},$ and the mass of each is 0.073 $\mathrm{kg}$ . One rod is held in place above the ground, while the other floats beneath it at a distance of $8.2 \times 10^{-3} \mathrm{m} .$ Determine the current in the rods.

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Bob Mckinnon

Numerade Educator

Problem 59

Two long, straight wires are separated by 0.120 $\mathrm{m} .$ The wires carry currents of 8.0 $\mathrm{A}$ in opposite directions, as the drawing indicates. Find the magnitude of the net magnetic field at the points labeled (a) $A$ and (b) $B$

Jarrad Pond

Numerade Educator

Problem 60

A long, straight wire carrying a current of 305 $\mathrm{A}$ is placed in a uniform magnetic field that has a magnitude of $7.00 \times 10^{-3}$ T. The wire is perpendicular to the field. Find a point in space where the net magnetic field is zero. Locate this point by specifying its perpendicular distance from the wire.

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Bob Mckinnon

Numerade Educator

Problem 61

mmh Two circular coils are concentric and lie in the same plane. The inner coil contains 140 turns of wire, has a radius of $0.015 \mathrm{m},$ and carries a current of 7.2 $\mathrm{A}$ . The outer coil contains 180 turns and has a radius of 0.023 $\mathrm{m}$ . What must be the magnitude and direction (relative to the current in the inner coil) of the current in the outer coil, so that the net magnetic field at the common center of the two coils is zero?

Jarrad Pond

Numerade Educator

Problem 62

A small compass is held horizontally, the center of its needle a distance of 0.280 $\mathrm{m}$ directly north of a long wire that is perpendicular to the earth's surface. When there is no current in the wire, the compass needle points due north, which is the direction of the horizontal component of the earth's magnetic field at that location. This component is parallel to the carth's surface. When the current in the wire is $25.0 \mathrm{A},$ the needle points $23.0^{\circ}$ east of north. (a) Does the current in the wire flow toward or away from the earth’s surface? (b) What is the magnitude of the horizontal component of the earth’s magnetic field at the location of the compass?

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Bob Mckinnon

Numerade Educator

Problem 63

Two infinitely long, straight wires are parallel and separated by a distance of one meter. They carry currents in the same direction. Wire 1 carries four times the current that wire 2 carries. On a line drawn perpendicular to both wires, locate the spot (relative to wire 1) where the net magnetic field is zero. Assume that wire 1 lies to the left of wire 2 and note that there are three regions to consider on this line: to the left of wire 1, between wire 1 and wire 2, and to the right of wire 2.

Jarrad Pond

Numerade Educator

Problem 64

The drawing shows two perpendicular, long, straight wires, both of which lie in the plane of the paper. The current in each of the wires is $I=5.6 \mathrm{A}$ . Find the magnitudes of the net magnetic fields at points $A$ and $B$

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Bob Mckinnon

Numerade Educator

Problem 65

ssm A piece of copper wire has a resistance per unit length of $5.90 \times 10^{-3} \Omega / \mathrm{m}$ . The wire is wound into a thin, flat coil of many turns that has a radius of 0.140 $\mathrm{m}$ . The ends of the wire are connected to a $12.0-\mathrm{V}$ battery. Find the magnetic field strength at the center of the coil.

Jarrad Pond

Numerade Educator

Problem 66

The drawing shows two wires that both carry the same current of $I=85.0 \mathrm{A}$ and are oriented perpendicular to the plane of the paper. The current in one wire is directed out of the paper, while the current in the other is directed into the paper. Find the magnitude and direction of the net magnitude and direction of the net magnetic field at point $P .$

BM

Bob Mckinnon

Numerade Educator

Problem 67

The drawing shows two long, straight wires that are suspended from a ceiling. The mass per unit length of each wire is 0.050 $\mathrm{kg} / \mathrm{m}$ . Each of the four strings suspending the wires has a length of 1.2 $\mathrm{m}$ . When the wires carry identical currents in opposite directions, the angle between the strings holding the two wires is $15^{\circ} .$ What is the current in each wire?

Jarrad Pond

Numerade Educator

Problem 68

Suppose that a uniform magnetic field is everywhere perpendicular to this page. The field points directly upward toward you. A circular path is drawn on the page. Use Ampère's law to show that there can be no net current passing through the circular surface.

BM

Bob Mckinnon

Numerade Educator

Problem 69

The wire in Figure 21.38 carries a current of 12 A. Suppose that a sccond long, straight wire is placed right next to this wire. The current in the second wire is 28 $\mathrm{A}$ . Use Ampère's law to find the magnitude of the magnetic field at a distance of $r=0.72 \mathrm{m}$ from the wires when the currents are $(\mathrm{a})$ in the same direction and $(\mathrm{b})$ in opposite directions.

Jarrad Pond

Numerade Educator

Problem 70

mmh A very long, hollow cylinder is formed by rolling up a thin sheet of copper. Electric charges flow along the copper sheet parallel to the axis of the cylinder. The arrangement is, in effect, a hollow tube of current I. Use Ampere's law to show that the magnetic field (a) is $\mu_{0} I /(2 \pi r)$ outside the cylinder at a distance $r$ from the axis and (b) is zero at any point within the hollow interior of the cylinder. (Hint: For closed paths, use circles perpendicular to and centered on the axis of the cylinder.

BM

Bob Mckinnon

Numerade Educator

Problem 71

ssm A long, cylindrical conductor is solid throughout and has a radius $R$ . Electric charges flow parallel to the axis of the cylinder and pass uniformly through the entire cross section. The arrangement is, in effect, a solid tube of current $I_{0} .$ The current per unit cross-sectional area (i.e., the current density ) is $I_{0} /\left(\pi R^{2}\right) .$ Use Ampere's law to show that the magnetic field inside the conductor at a distance $r$ from the axis is $\mu_{0} I_{0} r /\left(2 \pi R^{2}\right)$ . Hint: For a closed path, use $a$ circle of radius $r$ perpendicular to and centered on the axis. Note that the current through any surface is the area of the surface times the current density.)

Jarrad Pond

Numerade Educator

Problem 72

In a certain region, the carth's magnetic ficld has a magnitude of $5.4 \times 10^{-5} \mathrm{T}$ and is directed north at an angle of $58^{\circ}$ below the horizontal. An electrically charged bullet is fired north and $11^{\circ}$ above the horizontal, with a speed of 670 $\mathrm{m} / \mathrm{s}$ . The magnetic force on the bullet is $2.8 \times 10^{-10} \mathrm{N}$ directed due east. Determine the bullet's electric charge, including its algebraic sign $(+\text { or }-) .$

BM

Bob Mckinnon

Numerade Educator

Problem 73

ssm An electron is moving through a magnetic field whose magniude is $8.70 \times 10^{-4} \mathrm{T}$ . The clectron expericnces only a magnetic force and has an acceleration of magnitude $3.50 \times 10^{14} \mathrm{m} / \mathrm{s}^{2}$ . At a certain instant, it has a speed of $6.80 \times 10^{6} \mathrm{m} / \mathrm{s}$ . Determine the angle $\theta$ (less than $90^{\circ}$ ) between the electron's velocity and the magnetic field.

Jarrad Pond

Numerade Educator

Problem 74

A very long, straight wire carries a current of 0.12 A. This wire is tangent to a single-turn, circular wire loop that also carries a current. The directions of the currents are such that the net magnetic field at the center of the loop is zero. Both wires are insulated and have diameters that can be neglected. How much current is there in the loop?

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Bob Mckinnon

Numerade Educator

Problem 75

7ssm The maximum torque experienced by a coil in a 0.75 -T magnetic field is $8.4 \times 10^{-4} \mathrm{N}$ . The coil is circular and consists of only one turn. The current in the coil is 3.7 $\mathrm{A}$ . What is the length of the wire from which the coil is made?

Jarrad Pond

Numerade Educator

Problem 76

Multiple-Concept Example 7 discusses how problems like this one can be solved. A $+6.00 \mu \mathrm{C}$ charge is moving with a speed of $7.50 \times 10^{4} \mathrm{m} / \mathrm{s}$ parallel to a very long, straight wire. The wire is 5.00 $\mathrm{cm}$ from the charge and carries a current of 67.0 $\mathrm{A}$ in a direction opposite to that of the moving charge. Find the magnitude and direction of the force on the charge.

BM

Bob Mckinnon

Numerade Educator

Problem 77

The $x, y,$ and $z$ components of a magnetic field are $B_{x}=0.10 \mathrm{T}$ $B_{y}=0.15 \mathrm{T}$ , and $B_{z}=0.17 \mathrm{T} . \mathrm{A} 25-\mathrm{cm}$ wire is oriented along the $z$ axis and carries a current of 4.3 $\mathrm{A}$ . What is the magnitude of the magnetic force that acts on this wire?

Jarrad Pond

Numerade Educator

Problem 78

In a lightning bolt, a large amount of charge flows during a time of $1.8 \times 10^{-3}$ s. Assume that the bolt can be treated as a long, straight line of current. At a perpendicular distance of 27 $\mathrm{m}$ from the bolt, a magnetic field of $8.0 \times 10^{-5} \mathrm{T}$ is measured. How much charge has flowed during the lightning bolt? Ignore the carth's magnetic ficld.

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Bob Mckinnon

Numerade Educator

Problem 79

A charge is moving perpendicular to a magnetic field and experiences a force whose magnitude is $2.7 \times 10^{-3} \mathrm{N}$ . If this same charge were to move at the same speed and the angle between its velocity and the same magnetic field were $38^{\circ},$ what would be the magnitude of the magnetic force magnitude of the magnetic force that the charge would experience?

Jarrad Pond

Numerade Educator

Problem 80

The drawing shows four insulated wires overlapping one another, forming a square with 0.050 -m sides. All four wires are much longer than the sides of the square. The net magnetic field at the center of the square is 61$\mu \mathrm{T}$ . Calculate the current $I .$

BM

Bob Mckinnon

Numerade Educator

Problem 81

ssm A particle of charge $+7.3 \mu \mathrm{C}$ and mass $3.8 \times$ $10^{-8} \mathrm{kg}$ is traveling perpendicular to a $1.6-\mathrm{T}$ magnetic field, as the drawing shows. The speed of the particle is 44 $\mathrm{m} / \mathrm{s}$ . (a) What is the value of the angle $\theta,$ such that
the particle's subscquent path will intersect the $y$ axis at the greatest possible value of $y ?$ (b) Determine this value of $y .$

Jarrad Pond

Numerade Educator

Problem 82

A particle has a charge of $q=+5.60 \mu \mathrm{C}$ and is located at the coordinate origin. As the drawing shows, an electric field of $E_{x}=+245 \mathrm{N} / \mathrm{C}$ exists along the $+x$ axis, A magnetic field also exists, and its $x$ and $y$ components are $B_{x}=+1.80 \mathrm{T}$ and $B_{y}=+1.40 \mathrm{T}$ . Calculate the force (magnitude and direction) exerted on the particle by each of the three fields when it is $\quad$ stationary, (b) moving along the $+x$ axis at a speed of 375 $\mathrm{m} / \mathrm{s}$ , and $\quad$ (c) moving along the $+z$ axis at a speed of 375 $\mathrm{m} / \mathrm{s}$ .

BM

Bob Mckinnon

Numerade Educator

Problem 83

Two parallel rods are each 0.50 m in length. They are attached at their centers to either end of a spring (spring constant $=150 \mathrm{N} / \mathrm{m} )$ that is initially neither stretched nor compressed. When 950 $\mathrm{A}$ of current is in each rod in the same direction, the spring is observed to be compressed by 2.0 $\mathrm{cm}$ . Treat the rods as long, straight wires and find the separation between them when the current is present.

Jarrad Pond

Numerade Educator

Problem 84

A solenoid is formed by winding 25.0 $\mathrm{m}$ of insulated silver wire around a hollow cylinder. The turns are wound as closely as possible without overlapping, and the insulating coat on the wire is negligibly thin. When the solenoid is connected to an ideal (no internal resistance) $3.00-\mathrm{V}$ battery, the magnitude of the magnetic field inside the solenoid is found to be $6.48 \times 10^{-3} \mathrm{T}$ . Determine the radius of the wire. $(\text {Hint:$ Because the solenoid is closely coiled, the number of turns per unit length depends on the radius of the wire.)

BM

Bob Mckinnon

Numerade Educator

Problem 85

ssm A charge of $4.0 \times 10^{-6} \mathrm{C}$ is placed on a small conducting sphere that is located at the end of a thin insulating rod whose length is 0.20 $\mathrm{m}$ . The rod rotates with an angular speed of $\omega=150 \mathrm{rad} / \mathrm{s}$ about an axis that passes perpendicularly through its other end. Find the magnetic moment of the rotating charge. (Hint: The charge travels around a circle in a time equal to the period of the motion.)

Jarrad Pond

Numerade Educator